Puglia in March. Thoughts?

There is a direct train from Rome that goes thru Bari and down to Lecce. Since Lecce is your farthest south destination you might want to start in Rome and take the train to Lecce (about 6 hours). You can get around by public transit but a car is useful in Puglia. Lecce is more like a suburb than a city such as Rome or Florence in terms of driving difficulty, so there would be no problem staying in the historic center and parking at one of the lots just outside the walls.

We were there in Sept instead of spring. The retail shops, churches, museums closed around 1pm and didn't reopen until 4-6pm. Restaurants were open until around 2pm for lunch and then closed until 7-8pm. So most sightseeing was done in the morning. You might want to check what the hours would be in March but I suspect they would be similar.

Did you definitely want to include Rome? Did you price out flying in/out Bari and Brindisi? I did not find flying to either substantially more expensive than flying to Rome, so just making sure exhausted options. I would rather change planes (once) over having to transfer back and forth from Rome. It is worth the extra expense, to a point.
If the weather is nice, I would think you could find someone willing to take your money for a boat ride :)
All of those places will have open restaurants, etc. open. You only have to plan around the pausa. Check opening times for anything you want to see. For example, we figured the cathedral in Otranto would be open all day, but it was closed when we passed by.

It was hairy driving in and out of Lecce, but nothing I would especially go out of my way to avoid. (Of course we only drove out of Bari.) Everywhere else the driving is pretty easy. When you drive to Matera, be sure to map to the side of town where your hotel is, as there are multiple ways to approach and you'll want the easiest for avoiding ZTL.

Did you definitely want to include Rome?

I do not need to include Rome. I can get a direct overnight flight from Dulles to Rome so I was looking at the option of catching a connecting flight vs taking a train from there. After reading your comment, I looked on google flights and it looks like I can do a multi city from Dulles with a connection through FCO to Brindisi, then from Bari with a connecting flight through FCO straight back to Dulles. Not spending any nights in Rome gives me a few more nights in Puglia if I want them. I like that idea!

John, even driving in a smaller city makes me a little nervous. I studied signs and thought I was prepared for driving in France and Italy in the past, but I seemed to forget everything I thought I'd learned when put on the spot. Are you suggesting I can rent a car just outside Lecce? That might not be so bad, plus if I do that, idea 3) would allow me to drive up the coast to PaM, which I read is a lovely drive.

I don't mind that things shut down for a few hours in the afternoon, but hope they aren't closed altogether for the season in March.

Glad to hear it--I think that is the best way to maximize your time.
Once you get the flights sorted, you can group the places you want a car and vice versa. The places we wanted to visit from Lecce required a car, so as long as you can get to your designated parking spot, you'll be fine!
If you want to pick up or drop off from Bari city, there are places near the train station, and we drove right out of town with no problem.
You might find some restaurants closed, but otherwise I don't think you will find too much difference aside from fewer crowds. I never made it to Polignano, so I can't speak to that location, but I would have zero qualms about any other place you are staying. PaM and Trani will be quiet, but by no means shuttered.
Year-round, the pausa can be an issue in a small village or even a larger town--have your lunch plans sorted! We did scenic walks or drives during that time usually, or went to a museum.

I doubt if there are many major rental agencies in Locorotondo.
I would rent the car in Bari and return it either in Bari or Brindisi.

Rent at the airports to avoid driving INTO those cities.
I would not rent from a tiny local agency, but from one of the big names. (I use SIXT but there are others)

Boat trips in Polignano in March----could be difficult to find but I would not know for certain. I'm sure your hotels can help, or you can contact a few of the agencies that offer boat tours.

The only place I'm concerned about needing a car rather than train/bus travel is the area of Locorotondo, Alberobello, Martina Franca and Matera, and honestly, it looks like I could rent a bike to get to Alberobello and Martina Franca from Locorotondo, but getting to Matera has me stuck. I looked at private transfer and taxi prices yesterday and it was rather expensive, but train/bus takes hours since it goes into Bari first. I'll have to give that some more thought. It would be great to avoid a rental car altogether if I can.

Haven't done it myself because we had a car, but I know there is a train and bus combo to Matera from Bari. Jean from Idaho and others on this forum have taken it.